## Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators

HTML articles powered by AMS MathViewer

- by Steve Hofmann, Carlos Kenig, Svitlana Mayboroda and Jill Pipher PDF
- J. Amer. Math. Soc.
**28**(2015), 483-529 Request permission

## Abstract:

We consider divergence form elliptic operators $L= {-}\mathrm {div} A(x) \nabla$, defined in the half space $\mathbb {R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A(x)$ is bounded, measurable, uniformly elliptic, $t$-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation $Lu=0$, and we then combine these estimates with the method of “$\epsilon$-approximability” to show that $L$-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class $A_\infty$ with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in $L^p$, for some $p<\infty$). Previously, these results had been known only in the case $n=1$.## References

- Pascal Auscher,
*On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\Bbb R^n$ and related estimates*, Mem. Amer. Math. Soc.**186**(2007), no. 871, xviii+75. MR**2292385**, DOI 10.1090/memo/0871 - Pascal Auscher and Andreas Axelsson,
*Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I*, Invent. Math.**184**(2011), no. 1, 47–115. MR**2782252**, DOI 10.1007/s00222-010-0285-4 - M. Angeles Alfonseca, Pascal Auscher, Andreas Axelsson, Steve Hofmann, and Seick Kim,
*Analyticity of layer potentials and $L^2$ solvability of boundary value problems for divergence form elliptic equations with complex $L^\infty$ coefficients*, Adv. Math.**226**(2011), no. 5, 4533–4606. MR**2770458**, DOI 10.1016/j.aim.2010.12.014 - Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian,
*The solution of the Kato square root problem for second order elliptic operators on ${\Bbb R}^n$*, Ann. of Math. (2)**156**(2002), no. 2, 633–654. MR**1933726**, DOI 10.2307/3597201 - Pascal Auscher, Steve Hofmann, John L. Lewis, and Philippe Tchamitchian,
*Extrapolation of Carleson measures and the analyticity of Kato’s square-root operators*, Acta Math.**187**(2001), no. 2, 161–190. MR**1879847**, DOI 10.1007/BF02392615 - Pascal Auscher and Philippe Tchamitchian,
*Square root problem for divergence operators and related topics*, Astérisque**249**(1998), viii+172 (English, with English and French summaries). MR**1651262** - Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou,
*Asymptotic analysis for periodic structures*, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR**503330** - L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa,
*Boundary behavior of nonnegative solutions of elliptic operators in divergence form*, Indiana Univ. Math. J.**30**(1981), no. 4, 621–640. MR**620271**, DOI 10.1512/iumj.1981.30.30049 - R. R. Coifman and C. Fefferman,
*Weighted norm inequalities for maximal functions and singular integrals*, Studia Math.**51**(1974), 241–250. MR**358205**, DOI 10.4064/sm-51-3-241-250 - R. R. Coifman, Y. Meyer, and E. M. Stein,
*Some new function spaces and their applications to harmonic analysis*, J. Funct. Anal.**62**(1985), no. 2, 304–335. MR**791851**, DOI 10.1016/0022-1236(85)90007-2 - Björn E. J. Dahlberg,
*Approximation of harmonic functions*, Ann. Inst. Fourier (Grenoble)**30**(1980), no. 2, vi, 97–107 (English, with French summary). MR**584274** - Björn E. J. Dahlberg, David S. Jerison, and Carlos E. Kenig,
*Area integral estimates for elliptic differential operators with nonsmooth coefficients*, Ark. Mat.**22**(1984), no. 1, 97–108. MR**735881**, DOI 10.1007/BF02384374 - C. Fefferman and E. M. Stein,
*$H^{p}$ spaces of several variables*, Acta Math.**129**(1972), no. 3-4, 137–193. MR**447953**, DOI 10.1007/BF02392215 - John B. Garnett,
*Bounded analytic functions*, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**628971** - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190**, DOI 10.1007/978-3-642-61798-0 - Steve Hofmann,
*A local $Tb$ theorem for square functions*, Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 175–185. MR**2500492**, DOI 10.1090/pspum/079/2500492 - Hofmann S., Kenig C., Mayboroda S., and Pipher J.,
*The Regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients*. preprint., DOI 10.1007/s00208-014-1087-6 - Steve Hofmann, Michael Lacey, and Alan McIntosh,
*The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds*, Ann. of Math. (2)**156**(2002), no. 2, 623–631. MR**1933725**, DOI 10.2307/3597200 - Hofmann S., Mayboroda S., and Mourgoglou M.,
*$L^p$ and endpoint solvability results for divergence form elliptic equations with complex $L^{\infty }$ coefficients*. preprint. - Hofmann S., Mitrea M., and Morris A.,
*The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^\infty$ Coefficients*. preprint. - David S. Jerison and Carlos E. Kenig,
*The Dirichlet problem in nonsmooth domains*, Ann. of Math. (2)**113**(1981), no. 2, 367–382. MR**607897**, DOI 10.2307/2006988 - Tosio Kato,
*Perturbation theory for linear operators*, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR**0203473** - Carlos E. Kenig,
*Harmonic analysis techniques for second order elliptic boundary value problems*, CBMS Regional Conference Series in Mathematics, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR**1282720**, DOI 10.1090/cbms/083 - C. Kenig, H. Koch, J. Pipher, and T. Toro,
*A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations*, Adv. Math.**153**(2000), no. 2, 231–298. MR**1770930**, DOI 10.1006/aima.1999.1899 - Carlos E. Kenig, Fanghua Lin, and Zhongwei Shen,
*Homogenization of elliptic systems with Neumann boundary conditions*, J. Amer. Math. Soc.**26**(2013), no. 4, 901–937. MR**3073881**, DOI 10.1090/S0894-0347-2013-00769-9 - Carlos E. Kenig and Jill Pipher,
*The Neumann problem for elliptic equations with nonsmooth coefficients*, Invent. Math.**113**(1993), no. 3, 447–509. MR**1231834**, DOI 10.1007/BF01244315 - Carlos E. Kenig and Zhongwei Shen,
*Homogenization of elliptic boundary value problems in Lipschitz domains*, Math. Ann.**350**(2011), no. 4, 867–917. MR**2818717**, DOI 10.1007/s00208-010-0586-3 - Carlos E. Kenig and Zhongwei Shen,
*Layer potential methods for elliptic homogenization problems*, Comm. Pure Appl. Math.**64**(2011), no. 1, 1–44. MR**2743875**, DOI 10.1002/cpa.20343 - Andreas Rosén,
*Layer potentials beyond singular integral operators*, Publ. Mat.**57**(2013), no. 2, 429–454. MR**3114777**, DOI 10.5565/PUBLMAT_{5}7213_{0}8 - James Serrin and H. F. Weinberger,
*Isolated singularities of solutions of linear elliptic equations*, Amer. J. Math.**88**(1966), 258–272. MR**201815**, DOI 10.2307/2373060 - Varopoulos N.,
*A remark on BMO and bounded harmonic functions*(1970).

## Additional Information

**Steve Hofmann**- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 251819
- ORCID: 0000-0003-1110-6970
- Email: hofmanns@missouri.edu
**Carlos Kenig**- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois, 60637
- MR Author ID: 100230
- Email: cek@math.chicago.edu
**Svitlana Mayboroda**- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455
- MR Author ID: 739839
- Email: svitlana@math.umn.edu
**Jill Pipher**- Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
- MR Author ID: 237541
- Email: jpipher@math.brown.edu
- Received by editor(s): February 10, 2012
- Received by editor(s) in revised form: February 11, 2014
- Published electronically: May 21, 2014
- Additional Notes: Each of the authors was supported by the NSF

This work has been possible thanks to the support and hospitality of the*University of Chicago*, the*University of Minnesota*, the*University of Missouri*,*Brown University*, and the*BIRS Centre in Banff*(Canada). The authors would like to express their gratitude to these institutions. - © Copyright 2014 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**28**(2015), 483-529 - MSC (2010): Primary 42B99, 42B25, 35J25, 42B20
- DOI: https://doi.org/10.1090/S0894-0347-2014-00805-5
- MathSciNet review: 3300700